This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $mathbb{R}^n$ ($n>1$). Suppose the complement of $F$ is connected and let

$$A={xin mathbb{R}^n: dist(x, F)=delta}, $$ where $delta>0$ is fixed, and $dist$ is the Euclidean distance. If $F$ is unbounded with empty interior, can the complement of $A$ still have a bounded component?

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